Related papers: Solving the fully nonlinear Monge-Amp\`ere equatio…

This work introduces Probabilistic Kolmogorov-Arnold Network (P-KAN), a novel probabilistic extension of Kolmogorov-Arnold Networks (KANs) for time series forecasting. By replacing scalar weights with spline-based functional connections and…

Recent advancements in deep learning for image classification predominantly rely on convolutional neural networks (CNNs) or Transformer-based architectures. However, these models face notable challenges in medical imaging, particularly in…

We analyze the convergence of an iterative method for solving the nonlinear system resulting from a natural discretization of the Monge-Amp\`ere equation with $C^1$ conforming approximations. We make the assumption, supported by numerical…

This paper presents a comprehensive survey of 18 distinct polynomials and their potential applications in Kolmogorov-Arnold Network (KAN) models as an alternative to traditional spline-based methods. The polynomials are classified into…

We introduce the Kolmogorov-Arnold Network for Dynamics (KANDy) as a zero-depth, wide neural architecture capable of discovering governing equations in chaotic and complex dynamical systems. Building on the foundation of Kolmogorov-Arnold…

This study explores the application of Kolmogorov-Arnold Networks (KANs) in predicting nuclear binding energies, leveraging their ability to decompose complex multi-parameter systems into simpler univariate functions. By utilizing data from…

In this paper, the author studies quaternionic Monge-Amp\`ere equations and obtains the existence and uniqueness of the solutions to the Dirichlet problem for such equations without any restriction on domains. Our paper not only answers to…

Kolmogorov-Arnold Networks (KANs) have demonstrated an exceptional ability to learn complex functions on clean, low-dimensional data but struggle to maintain performance on noisy and imperfect real-world datasets. In contrast, conventional…

We present an adaptation of the MA-LBR scheme to the Monge-Amp{\`e}re equation with second boundary value condition, provided the target is a convex set. This yields a fast adaptive method to numerically solve the Optimal Transport problem…

We show that the discrete Sinkhorn algorithm - as applied in the setting of Optimal Transport on a compact manifold - converges to the solution of a fully non-linear parabolic PDE of Monge-Ampere type, in a large-scale limit. The latter…

This article presents an input convex neural network architecture using Kolmogorov-Arnold networks (ICKAN). Two specific networks are presented: the first is based on a low-order, linear-by-part, representation of functions, and a universal…

We firstly simulated disease dynamics by KAN (Kolmogorov-Arnold Networks) nearly 4 years ago, but the kernel functions in the edge include the exponential number of infected and discharged people and is also in line with the…

In this article, we introduce and study three numerical methods for the Dirichlet Monge Amp\`ere equation in two dimensions. The approaches consist in considering new equivalent problems. The latter are discretized by a wide stencil finite…

We introduce two hybridizable discontinuous Galerkin (HDG) methods for numerically solving the Monge-Ampere equation. The first HDG method is devised to solve the nonlinear elliptic Monge-Ampere equation by using Newton's method. The second…

Kolmogorov-Arnold Networks (KANs) uniquely combine high accuracy with interpretability, making them valuable for scientific modeling. However, it is unclear a priori how deep a network needs to be for any given task, and deeper KANs can be…

Inspired by the matching of supply to demand in logistical problems, the optimal transport (or Monge--Kantorovich) problem involves the matching of probability distributions defined over a geometric domain such as a surface or manifold. In…

Data science has emerged as fourth paradigm of scientific exploration. However many machine learning models operate as black boxes offering limited insight into the reasoning behind their predictions. This lack of transparency is one of the…

In this article, we report the results we obtained when investigating the numerical solution of some nonlinear eigenvalue problems for the Monge-Amp\`{e}re operator $v\rightarrow \det \mathbf{D}^2 v$. The methodology we employ relies on the…

We formulate an Alternating Direction Method of Mul-tipliers (ADMM) that systematically distributes the computations of any technique for optimizing pairwise functions, including non-submodular potentials. Such discrete functions are very…

The elliptic Monge-Amp\`ere equation is a fully nonlinear partial differential equation which has been the focus of increasing attention from the scientific computing community. Fast three dimensional solvers are needed, for example in…